In a communication system, an information value is often sent from one communication node (transmitter) to another communication node (receiver), via either a wired or a wireless channel. Due to the limited word length in a data packet where the information value is put in for the transmission, the information value is often quantized to a limited number of digits, or equivalently translated to a value range index that represents the value range which the information value falls into. For example, a general mapping table of N rows for such translation upon information value x is shown in Table 1. The information value to be quantized is assumed to be between a0 (inclusively) and aN (non-inclusively). The whole quantization range is denoted as [a0, aN). This quantization range is segmented into N contiguous value ranges by (N−1) values ai for 1≤i≤N−1, where ai for all 0≤i≤N are known to both transmitter and receiver, and meanwhile satisfies a0<a1<a2< . . . <aN−1<aN. Then if the value x falls into the value range [am, am+1) for some m satisfying 0≤m≤N−1, the corresponding value range index m is used to represent the quantized value x and is sent from the transmitter to the receiver.
TABLE 1General mapping table for quantization and transmissionValue range indexValue range where x falls into0a0 ≤ x < a11a1 ≤ x < a2......iai ≤ x < ai+1......N − 1aN−1 ≤ x < aN
On the receiver side that receives the value range index m, the receiver knows the value being quantized is within value range of [am, am+1), it can pick any value x, which satisfies am≤x≤am+1, as the recovery for the value x. The absolute difference |x−x| is the quantization error.
In order to reduce the quantization error |x−x|, a prior-art quantization method using scaling factor was proposed. Its principle is described as following: denote the quantization procedure as x=Q(x), where function Q(⋅) represents the quantization procedure mentioned above, including the translation from value x to a value range index and the recovery from value range index to the quantized value x. Assume the maximum quantization error with respect to quantization function Q(⋅), i.e., the quantization resolution of Q(⋅), is R, then with a positive scaling factor k, |k·x−Q(k·x)|<R holds, which further leads to
                x      -                        Q          ⁡                      (                          k              ·              x                        )                          k                  <            R      k        .  This is to say, the quantization resolution or the maximum quantization error can be reduced to be the original divided by k if the scaling factor k is used to scale the value x before the quantization on the transmitter side, and is used to de-scale (i.e., the operation performed at the receiver that is opposite to the scaling operation performed at the transmitter) the quantized value that is received by the receiver on the receiver side. However, the drawback of this solution is that the effective quantization range is also reduced to be the original divided by k, i.e., if the quantization function Q(⋅) allows the input to be within range [a0, aN), the quantization with scaling factor k only allows the value x to be within range of
      [                            a          0                k            ,                        a          N                k              )    .